VSCode: How do I open the documentation (ala `show hover`) in another pane?

Have a look at the Docs View extension. I think it does what you want:

VS Code extension that displays hover documentation in the sidebar or panel.

This is not possible in VSCode at the moment,
but there is a lot of chatter on GitHub about the Intellisense suggestion UI.

If this is a feature you're interested in, you can add your vote
(click thumbs up in the first post of the thread) or your voice here: https://github.com/microsoft/vscode/issues/33752

In the meanwhile, you can make the suggestion box transparent https://github.com/microsoft/vscode/issues/22439#issuecomment-600855615
so at least you can see the code it covers.

Also related, but not identical:
https://github.com/microsoft/vscode/issues/15667
https://github.com/microsoft/vscode/issues/30797